A tilted bilayer membrane, which consists of two thin films of transparent optically isotropic materials of different refractive indices, can function as a transmission quarter-wave retarder (QWR) at a high angle of incidence. A specific design using a cryolite-Si membrane in the infrared is presented, and its tolerances to small shifts of wavelength, incidence angle, and film thickness errors are discussed. Some designs provide a dual QWR in transmission and reflection. Such devices provide simple linear-to-circular (and circular-to-linear) polarization transformers. Bilayer eighth-wave retarders without diattenuation are also introduced

Azzam, R. M.A.

Mahmoud, Fadi A.

English

University of New Orleans

University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 2-2001 Tilted bilayer membranes as simple transmission quarter-wave retardation plates R. M.A. Azzam University of New Orleans, razzam@uno.edu Fadi A. Mahmoud Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs  Part of the Electrical and Electronics Commons, and the Optics Commons Recommended Citation R. M. A. Azzam and Fadi A. Mahmoud, "Tilted bilayer membranes as simple transmission quarter-wave retardation plates," J. Opt. Soc. Am. A 18, 421-425 (2001) This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact scholarworks@uno.edu. Tilted bilayer membranes as simple transmissionquarter-wave retardation platesR. M. A. AzzamDepartment of Electrical Engineering, University of New Orleans, Lakefront, New Orleans, Louisiana 70148Fadi A. MahmoudAdaptec, Inc., 691 South Milpitas Boulevard, Milpitas, California 95035Received June 5, 2000; accepted July 26, 2000; revised manuscript received August 22, 2000A tilted bilayer membrane, which consists of two thin films of transparent optically isotropic materials of dif-ferent refractive indices, can function as a transmission quarter-wave retarder (QWR) at a high angle of inci-dence. A specific design using a cryolite-Si membrane in the infrared is presented, and its tolerances to smallshifts of wavelength, incidence angle, and film thickness errors are discussed. Some designs provide a dualQWR in transmission and reflection. Such devices provide simple linear-to-circular (and circular-to-linear)polarization transformers. Bilayer eighth-wave retarders without diattenuation are also introduced. © 2001Optical Society of AmericaOCIS codes: 120.2130, 230.0230, 240.0310, 260.5430.1. INTRODUCTIONWave retarders are versatile optical devices for the con-trol and the analysis of polarized light with numerous ap-plications. The desired differential phase shift betweentwo orthogonal linear polarization components of incidentlight is often introduced by using linear birefringence in acrystalline plate or total internal reflection.1Coherent multiple-beam interference in a tiltedparallel-plane thin dielectric slab (membrane or pellicle)leads to a differential transmission phase shift D t be-tween the parallel ( p) and perpendicular (s) polariza-tions; hence such a slab of an optically isotropic material(Fig. 1) acts as the simplest possible wave retardationplate.2 The retardance D t vanishes at normal incidence,and hence the effect can be considered to arise from formbirefringence associated with the tilt of the slab. The up-per bound on D t is given by3D t max 5 arctan$@N 2 ~1/N !#/2%, (1)where N 5 N1 /N0 is the refractive index of the slab rela-tive to that of the surrounding medium (usually air orvacuum, in which case N0 5 1). The maximum retar-dance is attainable in the limit of grazing incidence, and,according to Eq. (1), its value is ,90°, so that a single ho-mogeneous slab cannot perform as a quarter-wave re-tarder (QWR). On the other hand, an eighth-wave re-tarder (EWR) is readily achieved with a pellicle ofrefractive index N . 1.4966 at an angle of incidence thatdecreases as N increases.In this paper it is shown that this basic limitation ofthe homogeneous thin slab is lifted if the slab is coatedwith a transparent thin film of a different refractive in-dex. Interference in such a bilayer membrane gives aQWR, albeit at a high angle of incidence. A specific in-frared (IR) QWR design using a cryolite (Na3AlF6) -Sipellicle is presented. A dual QWR in transmission andreflection is also realizable with certain bilayer mem-branes. The coated pellicle QWR is a simple (noncrystal-line) device that introduces no significant lateral or angu-lar deviation in the transmitted beam. It can bemounted in a rotary stage such that its surface normalprecesses in a cone around the incident light beam as anaxis. Thus a QWR of variable fast-axis azimuth is ob-tained. Reflection phase retarders that use periodic orquasiperiodic stacks of many layers have been describedby several authors.4–62. BASIC RELATIONSThe change in the state of polarization of light upontransmission through a bilayer membrane (Fig. 2) is de-termined by the ratio of complex-amplitude transmissioncoefficients for the p and s polarizations:r t 5 Tp /Ts 5 ~tan c t!exp~ jD t!. (2)Explicit expressions for the transmission (and reflection)coefficients of a bilayer (on a substrate) are available7;substitution of these expressions into Eq. (2) givesr t 5 a~1 1 bX1 1 cX2 1 dX1X2!/~1 1 eX1 1 fX2 1 gX1X2!, (3)in whicha 5 ~t01p /t01s!~t12p /t12s!~t20p /t20s!,b 5 r01sr12s , e 5 r01pr12p ,c 5 r12sr20s , f 5 r12pr20p ,d 5 r01sr20s , g 5 r01pr20p . (4)R. M. A. Azzam and F. A. Mahmoud Vol. 18, No. 2 /February 2001 /J. Opt. Soc. Am. A 4210740-3232/2001/020421-05$15.00 © 2001 Optical Society of Americarijn and tijn are the Fresnel reflection and transmissioncoefficients of the ij interface for the n polarization(n 5 p, s), andXi 5 exp~2j2pz i!, i 5 1, 2, (5)where z i is the normalized thickness of the ith film, i.e.,z i 5 di /Di , i 5 1, 2, (6)andDi 5 ~l/2!~Ni2 2 sin2 f!21/2, i 5 1, 2, (7)is the corresponding film thickness period, l is the wave-length of the incident light, w is the angle of incidence, Niis the refractive index of the ith film, and light is assumedto be incident from air or vacuum (N0 5 1).3. CONSIDERATION OF SEVERALMATERIAL SYSTEMSFor given refractive indices (N1 , N2) of the two films ofthe bilayer, the normalized film thicknesses (z1 , z2) areiterated on in search of the maximum transmission differ-ential phase shift D tmax at a given angle of incidence w byusing Eqs. (2)–(5). Figure 3 is a plot of uD tmaxu versus wfor five material systems S0, S1, S2, S3, and S4 whichcorrespond to (N1 , N2) 5 (4, 4), (1.38, 2.35), (1.35, 4),(1.25, 4), and (1.35, 3.42), respectively. S0 reduces tothe case of a single homogeneous film of Ge. S1 repre-sents a MgF2-ZnS system in the visible, and S2–S4 rep-resent a low-index (fluoride) film on a high-index Si or Gelayer in the IR.8 The following conclusions are readilydrawn from Fig. 3.1. The upper bound on uD tmaxu for the S0 referencecase (the homogeneous Ge layer) is 61.93°, in agreementwith Eq. (1).2. For a bilayer of two films of different refractive in-dices, the upper bound on uD tmaxu is 180° at grazing inci-dence ( f 5 90°).3. For a bilayer of two films of different refractive in-dices, uD tu 5 90° is possible at a high angle of incidence f(.80°). Therefore a bilayer membrane can be used as aQWR in transmission. To stay as far from grazing inci-dence as possible, the refractive-index contrast betweenthe two films of the bilayer should be as high as possible,which is attained in the IR spectral region.4. CRYOLITE-Si BILAYER TRANSMISSIONAND REFLECTION DUAL QUARTER-WAVE RETARDER AT f ˜ 82° ANDl˜10.6 mmAs a specific design we take (N1 , N2) 5 (1.35, 3.42),which corresponds to a cryolite-Si bilayer at the 10.6-mmCO2-laser wavelength, and f 5 82° angle of incidence.The normalized thickness z1 is assigned constant val-ues from 0 to 1 in steps of 0.1, and for each value of z1 , z2is scanned over the full range 0–1 in finer steps. Figure4 shows the resulting family of contours of r t in the com-plex plane, as calculated from Eq. (3). These contoursare circles because r t is a bilinear function of X2 for con-Fig. 1. Reflection and transmission of light by a tilted transpar-ent thin slab of an optically isotropic material (medium 1 ofthickness d1) in a transparent ambient (medium 0). p and s arethe linear polarizations parallel and perpendicular to the planeof incidence, respectively, and f is the angle of incidence.Fig. 2. Reflection and transmission of light by a tilted bilayermembrane of two optically isotropic thin films (media 1 and 2 ofthicknesses d1 and d2, respectively) in a transparent ambient(medium 0). p and s are the linear polarizations parallel andperpendicular to the plane of incidence, respectively, and f is theangle of incidence.Fig. 3. Maximum differential phase shift in light transmissionthrough a bilayer, uD tmaxu, versus angle of incidence w for five ma-terial systems S0, S1, S2, S3, and S4, which correspond to(N1 , N2) 5 (4, 4), (1.38, 2.35), (1.35, 4), (1.25, 4), and (1.35,3.42), respectively.422 J. Opt. Soc. Am. A/Vol. 18, No. 2 /February 2001 R. M. A. Azzam and F. A. Mahmoudstant X1 according to Eq. (3), and the bilinear transforma-tion maps the unit circle of X2 onto a circle9 of r t . Pointsof intersection of the circles with the imaginary axis inFig. 4 correspond to bilayers that produce a QWR intransmission, i.e., D t 5 690°.Multiple solutions (z1 , z2) for a transmission QWR, forboth D t 5 190° and D t 5 290°, are presented by theclosed contours in Fig. 5. Superimposed in Fig. 5 are thecorresponding solution loci for reflection phase shifts Dr5 190° and Dr 5 290° for the same bilayer. A surpriseis the intersection of the loci of transmission and reflec-tion QWR’s at the four points x, y, u, and v. These pointsrepresent four bilayers that function as a dual QWR(D t,r 5 690°) in transmission and reflection.We further consider the design marked x in Fig. 5.The normalized and least metric film thicknesses for thisFig. 4. Family of contours of the ratio of complex-amplitude pand s transmission coefficients r t in the complex plane for acryolite-Si bilayer with refractive indices (1.35, 3.42), as calcu-lated from Eq. (3), at f 5 82° angle of incidence. The normal-ized thickness of the first layer, z1 , assumes constant valuesmarked by each curve while the normalized thickness of the sec-ond film, z2 , is scanned.Fig. 5. Multiple solutions (z1 , z2) for bilayer transmissionquarter-wave retarders (QWR’s), for both D t 5 190° and D t5 290°, are presented by the closed contours. Superimposedare the corresponding solution loci for reflection phase shifts Dr5 190° and Dr 5 290° for the cryolite-Si bilayer at f 5 82°angle of incidence. The intersection points x, y, u, and v repre-sent bilayers that function as a dual QWR in transmission andreflection.Fig. 6. D t as a function of wavelength around l 5 10.6 mm forthe cryolite-Si pellicle QWR.Fig. 7. Effect on D t of errors of 65% of the cryolite film thick-ness d1 in the cryolite-Si pellicle QWR.R. M. A. Azzam and F. A. Mahmoud Vol. 18, No. 2 /February 2001 /J. Opt. Soc. Am. A 423bilayer are (0.7992, 0.1228) and (4.6164, 0.1988) mm, re-spectively. If the dispersion of the two film materials isneglected, the effect on D t of shifting the wavelengtharound the design wavelength l 5 10.6 mm is shown inFig. 6. The QWR is reasonably achromatic in the neigh-borhood of the design wavelength (udD tu , 1° betweenl 5 10.3 and 10.7 mm).Figures 7 and 8 show the effect on D t of errors of 65%of the thicknesses d1 and d2 , respectively, at the designwavelength l 5 10.6 mm and the angle of incidencef 5 82°. It is apparent that the layer thicknessesshould be controlled to within 62% to limit the phaseerror to ,1°.Figure 9 shows the effect on D t of shifting the angle ofincidence by 61°. As may be expected, operation at thelarge angle f 5 82° requires a collimated (e.g., laser)beam with milliradian beam divergence.Because of the unequal throughput for the p and s po-larization (diattenuation10) of the bilayer QWR, the elec-tric field vector of incident linearly polarized light must beoriented at the appropriate azimuth u (measured from theplane of incidence) to produce an output (transmitted) cir-cularly polarized beam. For design x we determine u5 59.862° , and the insertion loss for this coated-pelliclecircular polarizer under this condition is 0.3395.It is possible to design a bilayer EWR without diattenu-ation. For the same refractive indices (N1 , N2)5 (1.35, 3.42) and the same angle of incidence f5 82° , an EWR is achieved with normalized thicknesses(z1 , z2) 5 (0.1009, 0.9594). In this case D t 5 245°, andthe throughputs (intensity transmittances) for the p and spolarizations are the same and equal 84.5%. If two suchpellicles are used in series and with equal and oppositetilts (Fig. 10), a QWR with improved field of view is ob-tained. Other two-pellicle designs are possible, at lowerangles of incidence, in which the planes of incidence atthe two pellicles are made orthogonal to equalize the pand s net transmittances.5. CONCLUSIONPellicles are used traditionally as light, yet quite sturdybeam splitters.11 In this paper it is shown that, by ap-propriate design, a bilayer pellicle of optically isotropicmaterials can introduce 90° differential phase shift be-tween the p and s linear polarization components intransmission and hence can function as a simple QWRwith negligible lateral or angular shift. To lower the op-erating angle of incidence and remove diattenuation re-quire multilayer-coated pellicles. This extension of thepresent work falls outside the scope of this paper.Address correspondence to R. M. A. Azzam at the loca-tion on the title page or by e-mail, razzam@uno.edu.REFERENCES1. See, for example, D. Clarke and J. F. Grainger, PolarizedLight and Optical Measurement (Pergamon, Oxford, UK,1971).Fig. 8. Effect on D t of errors of 65% of the Si film thickness d2in the cryolite-Si pellicle QWR.Fig. 9. Effect on D t of shifting the angle of incidence f by 61around f 5 82° for the cryolite-Si pellicle QWR.Fig. 10. Two oppositely tilted pellicle eighth-wave retarders inseries function as a QWR.424 J. Opt. Soc. Am. A/Vol. 18, No. 2 /February 2001 R. M. A. Azzam and F. A. Mahmoud2. D. A. Holmes, ‘‘Wave optics theory of rotatory compensa-tors,’’ J. Opt. Soc. Am. 54, 1340–1347 (1964).3. R. M. A. Azzam, ‘‘Transmission ellipsometry on trans-parent unbacked or embedded thin films with applicationto soap films in air,’’ Appl. Opt. 30, 2801–2806(1991).4. W. H. Southwell, ‘‘Multilayer coating design achieving abroadband 90° phase shift,’’ Appl. Opt. 19, 2688–2692(1980).5. J. H. Apfel, ‘‘Phase retardance of periodic multilayer mir-rors,’’ Appl. Opt. 21, 733–738 (1982).6. M. M. K. Howlader and R. M. A. Azzam, ‘‘Periodic andquasiperiodic nonquarterwave multilayer coatings for 90-deg reflection phase retardance at 45-deg angle ofincidence,’’ Opt. Eng. 34, 869–875 (1995).7. R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolarized Light (North-Holland, Amsterdam, 1987).8. E. D. Palik, ed., Handbook of Optical Constants of Solids(Academic, New York, 1985).9. A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972).10. R. A. Chipman, ‘‘Polarization analysis of optical systems,’’Opt. Eng. 28, 90–99 (1989).11. J. A. Dobrowolski, ‘‘Optical properties of films and coat-ings,’’ in Handbook of Optics, M. Bass, editor-in-chief(McGraw-Hill, New York, 1995), Vol. I, Chap. 42.R. M. A. Azzam and F. A. Mahmoud Vol. 18, No. 2 /February 2001 /J. Opt. Soc. Am. A 425

Tilted bilayer membranes as simple transmission quarter-wave retardation plates

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Tilted bilayer membranes as simple transmission quarter-wave retardation plates

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